Method for estimating the position and speed of an actuator body in an electromagnetic actuator for controlling the valve of an engine

ABSTRACT

Method for estimating the position and the speed of an actuator body in an electromagnetic actuator for controlling a valve of an engine, according to which, starting from a known value for the position and a first moment, a value is calculated at the first moment of the magnetic flux passing through a magnetic circuit constituted by an electromagnet and by the actuator body, the value for the speed at the first moment is estimated as a function of the magnetic flux and the position at the first moment, and the value is calculated at a second moment following the first moment and separated from said first moment by an interval of time determined by adding to the value of the position at the first moment the product of the speed at the first moment for the interval of time.

[0001] The present invention relates to a method for estimating theposition and speed of an actuator body in an electromagnetic actuatorfor controlling a valve of an engine.

BACKGROUND OF THE INVENTION

[0002] As is known, experiments are currently being conducted oninternal combustion engines of the type described in Italian patentapplication BO99A000443 filed on 4, Aug. 1999, in which the intake andexhaust valves are set in motion by electromagnetic actuators. Suchelectromagnetic actuators have undoubted advantages, in that they makeit possible to control each valve according to a law optimised for eachoperating condition of the engine, whereas traditional mechanicalactuators (typically camshafts) require the definition of a valve liftprofile that represents an acceptable compromise for all possibleoperating conditions of the engine.

[0003] An electromagnetic actuator for a valve of an internal combustionengine of the type described above normally comprises at least oneelectromagnet capable of displacing an actuator body made offerromagnetic material and mechanically connected to the stem of therespective valve. In order to apply a particular law of motion to thevalve, a control unit drives the electromagnet with a time-variablecurrent in order to displace the actuator body in a suitable manner.

[0004] From experimental testing it has been observed that, in order toachieve relatively high precision in controlling the valve it isnecessary to have feedback control of the position of the actuator body;it is therefore necessary to have an accurate—and substantiallyreal-time—reading of the position of said actuator body at any time. Inorder to achieve high performance levels from the feedback control it isfurthermore preferable also to have an accurate—and substantiallyreal-time—reading of the speed of the actuator body at any time.

[0005] In electromagnetic actuators of the type described above, theposition of the actuator body is read by a laser sensor, which is,however, expensive, delicate and difficult to calibrate and is thereforeunsuitable for use in mass production. Furthermore, the speed of theactuator body is estimated in a time-derivation operation on theposition of said actuator body at any time. However, such an operationsupplies a relatively inaccurate result in that it tends to amplify thenoise present when measuring the position of the actuator body.

SUMMARY OF THE INVENTION

[0006] The aim of the present invention is to provide a method forestimating the position and speed of an actuator body in anelectromagnetic actuator for controlling a valve of an engine, whichdoes not have the drawbacks described and, in particular, is easy andeconomical to operate.

[0007] According to the present invention a method is provided forestimating the position and speed of an actuator body in anelectromagnetic actuator for controlling a valve of an engine as claimedin claim 1.

BRIEF DESCRIPTION OF THE DRAWINGS

[0008] The present invention will now be described with reference to theattached drawings, which illustrate a few non-exhaustive embodimentsthereof, in which:

[0009]FIG. 1 is a diagrammatic view, in side elevation and in partialsection, of a valve of an engine and of a corresponding electromagneticactuator operating according to the method that is the subject-matter ofthe present invention;

[0010]FIG. 2 is a diagrammatic view of a control unit for the device inFIG. 1;

[0011]FIG. 3 illustrates diagrammatically a part of the control unit ofFIG. 2; and

[0012]FIG. 4 illustrates a circuit diagram of a detail of FIG. 3.

DETAILED DESCRIPTION OF THE INVENTION

[0013] In FIG. 1 an electromagnetic actuator 1 (of the type described inItalian patent application BO99A000443 filed on 4, Aug. 1999) isindicated as a whole by the reference number 1, coupled to an intake orexhaust valve 2 of an internal combustion engine of a known type fordisplacing said valve 2 along a longitudinal axis 3 of the valve betweena closed position (known and not illustrated) and a maximally openposition (known and not illustrated).

[0014] The electromagnetic actuator 1 comprises a swinging arm 4 made atleast partly of ferromagnetic material, which has a first end hinged toa support 5 so as to be able to oscillate about an axis 6 of rotationperpendicular to the longitudinal axis 3 of the valve 2, and a secondend connected by a connector 7 to an upper end of the valve 2. Theelectromagnetic actuator 1 also comprises two electromagnets 8 carriedin a fixed position by the support 5 so as to be arranged on oppositesides of the swinging arm 4, and a spring 9 coupled to the valve 2 andcapable of holding the swinging arm 4 in an intermediate position(illustrated in FIG. 1) in which said swinging arm 4 is equidistant fromthe pole pieces 10 of the two electromagnets 8.

[0015] In use, the electromagnets 8 are controlled by a control unit 11so as to exert alternately or simultaneously a force of attraction ofmagnetic origin on the swinging arm 4 in order to make it rotate aboutthe axis 6 of rotation, thereby displacing the valve 2 along therespective longitudinal axis 3 and between the aforementioned maximallyopen and closed positions (not illustrated). In particular, the valve 2is in the aforementioned closed position (not illustrated) when theswinging arm 4 is abutting against the upper electromagnet 8, is in theaforementioned maximally open position (not illustrated) when theswinging arm 4 is abutting against the lower electromagnet 8, and is ina partly open position when the two electromagnets 8 both have powershut off and the swinging arm 4 is in the aforementioned intermediateposition (illustrated in FIG. 1) by the effect of the force exerted bythe spring 9.

[0016] The control unit 11 provides feedback control, in a substantiallyknown manner, for the position of the swinging arm 4, i.e. the positionof the valve 2, on the basis of the conditioning of engine function. Inparticular, according to the illustration in FIG. 2, the control unit 11comprises a reference generation block 12, a calculation block 13, adriving block 14 capable of supplying the electromagnets 8 withtime-variable current, and an estimation block 15 capable ofestimating—substantially in real time—the position x(t) and, wherenecessary, the speed v(t) of the swinging arm 4.

[0017] In use, the reference generation block 12 receives as inputs aplurality of parameters indicating the operating conditions of theengine (for example the load, the engine speed, the position of thethrottle body, the angular position of the drive shaft, the temperatureof the coolant) and supplies the calculation block 13 with a targetvalue x_(R)(t) (i.e. a desired value) for the position of the swingingarm 4 (and therefore the valve 2).

[0018] On the basis of the target value x_(R)(t) for the position of theswinging arm 4 and on the basis of the estimated value x(t) of theposition of the swinging arm 4 received from the estimation block 15,the calculation block 13 prepares and sends to the driving block 14 acontrol signal z(t) for driving the electromagnets 8. In a preferredembodiment, the calculation block 13 prepares the control signal z(t)also on the basis of an estimated value v(t) for the speed of theswinging arm 4 received from the estimation block 15.

[0019] According another embodiment, not illustrated, the referencegeneration block 12 supplies the calculation block 13 with either atarget value x_(R)(t) for the position of the swinging arm 4, or atarget valve x_(R)(t) for the speed of the swinging arm 4.

[0020] As illustrated in FIG. 3, the driving block 14 supplies power tothe two electromagnets 8, each of which is composed of a respectivemagnetic core 16 coupled to a corresponding coil 17, for displacing theswinging arm 4 on the basis of the commands received from thecalculation block 13. The estimation block 15 reads the values, as shownin detail below, either from the driving block 14, or from the twoelectromagnets 8, in order to calculate an estimated value x(t) for theposition and an estimated value v(t) for the speed of the swinging arm4.

[0021] The swinging arm 4 is arranged between the pole pieces 10 of thetwo electromagnets 8, which are carried by the support 5 in a fixedposition and at a fixed distance D from one another, and therefore theestimated value x(t) of the position of the swinging arm 4 can beobtained directly with a simple operation of algebraic addition from anestimated value d(t) of the distance between a given point on theswinging arm 4 and a corresponding point on the one of the twoelectromagnets 8. By analogy, the estimated value v(t) for the speed ofthe swinging arm 4 can be obtained directly from an estimated value forthe speed existing between a given point on the swinging arm 4 and acorresponding point on one of the two electromagnets 8.

[0022] In order to calculate the value x(t) the estimation block 15calculates the two estimated values d₁(t), d₂(t) for the distancebetween a given point on the swinging arm 4 and a corresponding point onone of the two electromagnets 8; from the two estimated values d₁(t),d₂(t), the estimation block 15 obtains two values x₁(t), x₂(t), whichgenerally differ from one another because of measuring errors and noise.According to a preferred embodiment, the estimation block 15 takes anaverage of the two values x₁(t), x₂(t), weighted if necessary on thebasis of the accuracy attributed to each value x(t). By analogy, inorder to calculate the value v(t) the estimation block 15 calculates thetwo estimated values for speed existing between a given point on theswinging arm 4 and a corresponding point on one of the twoelectromagnets 8; from the two estimated values for speed, theestimation block 15 obtains two values v₁(t), v₂(t), which generallydiffer from one another because of measuring errors and noise. Accordingto a preferred embodiment, the estimation block 15 takes an average ofthe two values v₁(t), v₂(t), weighted if necessary on the basis of theaccuracy attributed to each value v(t).

[0023] With particular reference to FIG. 4, which illustrates a singleelectromagnet 8, a description is given below of the method used by theestimation block 15 for calculating an estimated value d(t) for thedistance between a given point on the swinging arm 4 and a correspondingpoint on the electromagnet 8, and for calculating an estimated value forthe speed existing between a given point on the swinging arm 4 and acorresponding point on the electromagnet 8.

[0024] In use, when the driving block 14 applies a voltage v(t) variableover time to the terminals of the coil 17 of the electromagnet 8, acurrent i(t) passes through said coil 17, consequently generating a fluxφ(t) over a magnetic circuit 18 coupled to the coil 17. In particular,the magnetic circuit 18 coupled to the coil 17 is composed of the core16 of ferromagnetic material of the electromagnet 8, the swinging arm 4made of ferromagnetic material and the air gap 19 existing between thecore 16 and the swinging arm 4.

[0025] The magnetic circuit 18 has a total reluctance R defined by thesum of the reluctance of iron R_(fe) and the reluctance of the air gapR₀; the value for the flux φ(t) circulating over the magnetic circuit 18is connected to the value of the current i(t) circulating within thecoil 17 by the following relationship (in which N is the number of turnsin the coil 17):

N*i(t)=R*φ(t)

R=R _(fe) +R ₀

[0026] In general the value for total reluctance R depends both on theposition x(t) of the swinging arm 4 (i.e. the breadth of the air gap 19,which is equal, except for a constant, to the position x(t) of theswinging arm 4), and on the assumed value for flux φ(t) . Except fornegligible errors (i.e. those of a first approximation) it can bedetermined that the value for reluctance of iron R_(fe) depends solelyon the assumed value for flux φ(t), while the value for reluctance ofthe air gap R₀ depends solely on the position x(t), i.e.

R(x(t), φ(t))=R _(fe)(φ(t))+R₀(x(t))

N*i(t)=R(x(t), φ(t)) *φ(t)

N*i(t)=R_(fe)(φ(t))*φ(t)+R ₀(x(t))* φ(t)

[0027] By solving the last equation given above with regard to R₀(x(t)),it is possible to obtain the value of the reluctance of the air gap R₀knowing the value of the current i(t), which value can easily bemeasured by an ammeter 20, knowing the value of N (fixed and dependenton the structural properties of the coil 17), knowing the value of theflux φ(t), and knowing the relationship between the reluctance of theiron (R_(fe) and the flux φ (known from the structural properties of themagnetic circuit 18 and the magnetic properties of the material used, oreasily determined by experimental tests).

[0028] The relationship between reluctance at the air gap R₀ and theposition x can be obtained relatively simply by analysing the propertiesof the magnetic circuit 18 (an example of a model of the behaviour ofthe air gap 19 is represented by the equation given below). Once therelationship between reluctance at the air gap R₀ and the position x isknown, the position x can be obtained from the reluctance at the air gapR₀ by applying the inverse relationship (applicable either by using theexact equation or by applying approximate numerical calculationmethods). The above statements can be summarised in the followingrelationships (where H_(fe)(φ(t))=R_(fe)(φ((t))*φ(t)): $\begin{matrix}{{R_{o}\left( {x(t)} \right)} = \frac{{N \cdot {i(t)}} - {H_{fe}\left( {\phi (t)} \right)}}{\phi (t)}} \\{{R_{o}\left( {x(t)} \right)} = {{K_{1}\left\lbrack {1 - ^{{- k_{2}}{x{(t)}}} + {k_{3} \cdot {x(t)}}} \right\rbrack} + K_{0}}} \\{{x(t)} = {{R_{0}^{- 1}\left( {R_{o}\left( {x(t)} \right)} \right)} = {R_{0}^{- 1}\left( \frac{{N \cdot {i(t)}} - {H_{fe}\left( {\phi (t)} \right)}}{\phi (t)} \right)}}}\end{matrix}$

[0029] The constants K₀, K₁, K₂, K₃ are constants that can be obtainedin experimental tests by using a series of measurements on the magneticcircuit 18.

[0030] From the above, it is clear that if the flux φ(t) can be measuredit is possible to calculate relatively easily the position x(t) of theswinging arm 4.

[0031] In a first embodiment, the flux φ(t) can be calculated bymeasuring the current i(t) that circulates through the coil 17 by usingthe ammeter 20 of a known type, measuring the voltage v(t) applied tothe terminals of the coil 17 by using a voltmeter 21 of known type, andknowing the value for resistance RES of the coil 17 (a value that iseasy to measure) . This method of measuring the flux φ(t) is based onthe following relationships: $\begin{matrix}{\frac{{\phi (t)}}{t} = {{v(t)} - {{RES} \cdot {i(t)}}}} \\{{\phi (T)} = {{\int_{0}^{T}{\left( {{v(t)} - {{RES} \cdot {i(t)}}} \right){t}}} + {\phi (0)}}}\end{matrix}$

[0032] The conventional moment 0 is chosen so as to find out accuratelythe value of the flux φ(0) at said moment 0; in particular, the moment 0is normally chosen within a period of time in which no current isflowing through the coil 17 and, therefore, the flux φ is substantiallyzero (the effect of any residual magnetisation is negligible), or themoment 0 is chosen according to a given position of the swinging arm 4(typically when the swinging arm 4 is abutting against the pole pieces10 of the electromagnet 8), in correspondence with which the value ofthe position x is known and therefore the value of the flux φ is known.

[0033] The method stated above for calculating the flux φ(t) is fairlyaccurate and fast (i.e. involving no delay); however, said method has afew problems, owing to the fact that the voltage v(t) applied to theterminals of the coil 17 is normally generated by a switching amplifierincorporated into the driving block 14 and therefore varies continuouslybetween three values (+V_(supply), 0, −V_(supply)) the continuousvariation (with very abrupt rises and falls) of the voltage v(t) makesit very difficult to measure said voltage v(t) accurately and quicklyand, consequently, to estimate the flux φ(t). In order to increaseaccuracy, the reading signal of the voltmeter 21 can be filtered inorder to attenuate the high frequencies, but such filtering inevitablyintroduces a delay into the measuring process.

[0034] In another embodiment, the magnetic coil 16 is coupled to anauxiliary turn (or coil) 22, to the terminals of which another voltmeter23 is connected; since the terminals of the turn 22 are substantiallyopen (the internal resistance of the voltmeter 23 is so high as to beregarded as infinite without thereby introducing appreciable errors), nocurrent flows through the turn 22 and the voltage v_(aux)(t) at itsterminals depends solely on the time derivative of the flux φ(t), fromwhich the flux can be deduced by means of a operation of integration (asconcerns the value φ(0), see the considerations stated above):$\begin{matrix}{\frac{{\phi (t)}}{t} = {v_{aus}(t)}} \\{{\phi (T)} = {{\int_{0}^{T}{{v_{aus}(t)}{t}}} + {\phi (0)}}}\end{matrix}$

[0035] From experimental tests it has been demonstrated that, incontrast to the voltage v(t) at the terminals of the coil 17, thevoltage v_(aux)(t) is substantially direct because of the effect ofmagnetic inertia (particular the stray currents induced in the iron) ofthe magnetic circuit 18 that damp the effects of the abrupt variationsin the voltage v(t) . In other words, the iron part of the magneticcircuit 18 has a low-pass filter effect that damps the abrupt variationsin the voltage v(t) and makes the voltage v_(aux)(t) substantiallydirect without introducing delays in measurement.

[0036] As stated above it is clear that by using the reading of thevoltage v_(aux)(t) of the auxiliary turn 22, calculation of the value ofthe flux φ(t) is more accurate and/or faster than using the reading ofthe voltage v(t) at the heads of the coil 17.

[0037] As well as for estimating the position x(t) of the swinging arm4, measurement of the flux φ(t) can be used by the control unit 11 forverifying the value of the force f(t) of attraction exerted by theelectromagnet 8 on the swinging arm 4, where: $\begin{matrix}{{f(t)} = {{- \frac{1}{2}} \cdot \frac{\partial{R\left( {{x(t)},{\phi (t)}} \right)}}{\partial x} \cdot {\phi^{2}(t)}}} \\{{f(t)} = {{- \frac{1}{2}} \cdot \frac{\partial{R_{0}\left( {x(t)} \right)}}{\partial x} \cdot {\phi^{2}(t)}}}\end{matrix}$

[0038] On the basis of the value of the position x(t) of the swingingarm 4, it is possible to calculate the value of the speed v(t) of theswinging arm 4 by using a simple time-derivative operation on theposition x(t); however, the value for speed v(t) obtained with such aderivation operation has much interference, since, as is known, thederivation operation markedly amplifies high-frequency interference. Toreduce the incidence of such interference it is necessary to carry outthe filtering operations with low-pass type filters which, however,introduce inevitable delays in estimating the value of the speed v(t).

[0039] According to another embodiment, both the position x(t) and thespeed v(t) can be calculated by using a process of calculation of theiterative type; this process is based on the equation (described above):

i(t)=R₀(x(t))*φ+H_(fe)(φ(t))

[0040] deriving said equation with respect to time and applying the lawsof partial derivation gives the equation: $\begin{matrix}{\frac{{i(t)}}{t} = {{\frac{\partial{R_{0}\left( {x(t)} \right)}}{\partial x} \cdot \frac{{x(t)}}{t} \cdot {\phi (t)}} + {{R_{0}\left( {x(t)} \right)} \cdot \frac{{\phi (t)}}{t}} + {\frac{\partial{H_{fe}\left( {\phi (t)} \right)}}{\partial\phi} \cdot \frac{{\phi (t)}}{t}}}} \\{\frac{{x(t)}}{t} = \frac{\frac{{i(t)}}{t} - {{R_{0}\left( {x(t)} \right)} \cdot \frac{{\phi (t)}}{t}} - {\frac{\partial{H_{fe}\left( {\phi (t)} \right)}}{\partial\phi} \cdot \frac{{\phi (t)}}{t}}}{\frac{\partial{R_{0}\left( {x(t)} \right)}}{\partial x} \cdot {\phi (t)}}}\end{matrix}$

[0041] reading from left to right it can be seen that: the timederivative of the current i(t) can be calculated easily by deriving themeasurement signal of the ammeter 20 (this signal is generally veryclean (i.e. free from noise) and free from abrupt variations and,therefore, can be time-derived with no particular problems);

[0042] the partial derivative of the reluctance R₀ of the air gap 19with respect to the position x can be calculated as a simple derivationof the equation R₀=R₀(x) described above;

[0043] the time derivative of the position x(t) is the speed v(t);

[0044] the flux φ(t) can be calculated by using one of the two methodsdescribed above;

[0045] the reluctance R₀ of the air gap 19 can easily be calculated fromthe equation R₀=R₀(x) described above if the value of the position x isknown;

[0046] the partial derivative of the quantity of ampere-turns Hfe of theiron with respect to the flux φ can be obtained easily if the structuralproperties of the magnetic circuit 18 are known; and

[0047] the time derivative of the flux φ(t) can be calculated with oneof the two methods described above.

[0048] Assuming that we are starting from a conventional moment t=0 inwhich both the value of the flux φ and the value of the position x areknown (as described above, this moment 0 is normally chosen at themoment in which the swinging arm 4 is in a given position, typicallyabutting against the pole pieces 10 of the electromagnet 8).

[0049] Starting from the moment t=0, the value of the reluctance R₀ ofthe air gap 19 is calculated at the moment t=0 using the value of theposition x(0) at the moment 0; inserting this value into the lastequation described above (and previously also calculating the othervalues in this equation by the method indicated earlier), it is possibleto calculate very easily the value of the speed v(0) at the moment t=0.

[0050] If a substantially negligible error is committed, it may beassumed that the speed v remains substantially constant for a period oftime dt (of a very small amplitude and depending on the desiredaccuracy); on the basis of this hypothesis, after the time dt, theposition x(0+dt) at the moment 0+dt will be:

x(0+dt)=x(0+v(0)*dt

[0051] in this way the value of the position x(0+dt) at the moment 0+dtis calculated, and the operations described above are repeated until thevalue of the speed v(0+dt) at the moment 0+dt is determined, and so on.

[0052] The method described above has the merit of supplying accuratelyand quickly either the value of the position x, or the value of thespeed v.

[0053] In the description given above two methods have been provided forestimating the time derivative of the flux φ(t) (hence the value of theflux φ(t) can be calculated), and two methods for calculating theposition x(t) and the speed v(t). According to one embodiment a choiceis made to use only one method for calculating the time derivative ofthe flux φ(t) and one method for calculating the position x(t) and thespeed v(t). According to another embodiment the choice is made to useboth methods for calculating the time derivative of the flux φ(t) and/orboth the methods for calculating the position x(t) and the speed v(t),and to use an average (weighted if necessary with respect to theestimated accuracy) of the results of the two methods used, or to useone result in order to verify the other (if there is a notableinconsistency between the two results it is likely that an error inestimating will be verified).

[0054] Finally, it is useful to observe that the methods described abovefor estimating the position x(t) and the speed v(t) can be used onlywhen there is a current passing through the coil 17 of an electromagnet8. For this reason, as described above, the estimation block 15 workswith both the electromagnets 8, so as to use the estimation performedwith one electromagnet 8 when the other is switched off. When both theelectromagnets are on, the estimation block 15 performs an average ofthe two values x(t) calculated with both electromagnets 8, weighted ifnecessary on the basis of the accuracy attributed to each value x(t)(generally the estimation of the position x made with respect to anelectromagnet 8 is more accurate when the swinging arm 4 is relativelyclose to the pole piece 10 of said electromagnet 8.

1. Method for estimating the position (x) and the speed (v) of anactuator body (4) in an electromagnetic actuator (1) for controlling avalve (2) of an engine; the actuator body (4) being made at least partlyof ferromagnetic material and being displaced towards at least oneelectromagnet (8) through the effect of the force of magnetic attractiongenerated by said electromagnet (8); the method being characterised bythe fact that starting from a known value of the position (x) and afirst moment (T1), a value is calculated at the first moment (T1) of themagnetic flux (p) passing through a magnetic circuit (18) constituted bythe electromagnet (8) and by the actuator body (4), the value of thespeed (v) at the first moment (T1) is estimated as a function of themagnetic flux (φ) and the position (x) at the first moment (T1), and thevalue is calculated at a second moment (T2) following the first moment(T1) and separated from said first moment (T1) by an interval of time(dt) determined by adding to the value of the position (x) at the firstmoment (T1) the product of the speed (v) at the first moment (T1) forthe interval of time (dt).
 2. Method according to claim 1, in which saidelectromagnet (8) defines, together with said actuator body (4), amagnetic circuit (18) influenced by a magnetic flux (φ) produced by acoil (17) through which an electric current (i) passes; said magneticcircuit (18) having a total reluctance (R), which is assumed to becomposed of the sum of a first reluctance (R₀) arising from an air gap(19) in the magnetic circuit (18) and a second reluctance (R_(fe))arising from the part of the magnetic circuit (18) made of ferromagneticmaterial (4, 16); the first reluctance (R₀) depending on the structuralproperties of the magnetic circuit (18) and on the value of the position(x), while the second reluctance (R_(fe)) depending on the structuralproperties of the magnetic circuit (18) and on the value of the magneticflux (φ) passing through the magnetic circuit (18).
 3. Method accordingto claim 2, in which the value for said first reluctance (R₀) and thevalue for said position (x) are connected by the following equation: R₀(x(t))=K ₁[1−e ^(−k) ^(₂) ^(x(t)) +k ₃ ·x(t)]+K ₀ in which R₀ is saidfirst reluctance (R₀), x(t) is said position (x) and K₀, K₁, K₂, K₃ arefour constants.
 4. Method according to claim 2, in which therelationship between the speed (v), the magnetic flux (φ) and theposition (x) is supplied by the following equation: $\begin{matrix}{\frac{{i(t)}}{t} = {{\frac{\partial{R_{0}\left( {x(t)} \right)}}{\partial x} \cdot \frac{{x(t)}}{t} \cdot {\phi (t)}} + {{R_{0}\left( {x(t)} \right)} \cdot \frac{{\phi (t)}}{t}} + {\frac{\partial{H_{fe}\left( {\phi (t)} \right)}}{\partial\phi} \cdot \frac{{\phi (t)}}{t}}}} \\{\frac{{x(t)}}{t} = \frac{\frac{{i(t)}}{t} - {{R_{0}\left( {x(t)} \right)} \cdot \frac{{\phi (t)}}{t}} - {\frac{\partial{H_{fe}\left( {\phi (t)} \right)}}{\partial\phi} \cdot \frac{{\phi (t)}}{t}}}{\frac{\partial{R_{0}\left( {x(t)} \right)}}{\partial x} \cdot {\phi (t)}}}\end{matrix}$

in which i is the electric current (i) circulating within the coil (17),R₀ is said first reluctance (R₀), x is the position (x) of the actuatorbody (4), φ is the magnetic flux (φ) and H_(fe) is the quantity ofampere-turns acted on by the iron part (4, 16) of the magnetic circuit(18).
 5. Method according to claim 1, in which the value of the magneticflux (φ) is estimated by measuring the value assumed from some electricparameters (i, v; va) of an electric circuit (17; 22) coupled to themagnetic circuit (18), calculating the time derivative of the magneticflux (φ) as a linear combination of the values of the electricalparameters (i, v; va), and integrating in time the derivative of themagnetic flux (φ).
 6. Method according to claim 5, in which the current(i) circulating through a coil (17) of the electromagnet (8) and thevoltage (v) applied to the terminals of said coil (17) are measured; thetime derivative of the magnetic flux (φ) and the magnetic flux (φ)itself being calculated by applying the following formulae:$\begin{matrix}{\frac{{\phi (t)}}{t} = {{v(t)} - {{RES} \cdot {i(t)}}}} \\{{\phi (T)} = {{\int_{0}^{T}{\left( {{v(t)} - {{RES} \cdot {i(t)}}} \right){t}}} + {\phi (0)}}}\end{matrix}$

in which φ is the magnetic flux (φ), v is the voltage (v) applied to theterminals of the coil (17), RES is the resistance of the coil (17) and iis the current (i) circulating through the coil (17).
 7. Methodaccording to claim 6, in which the voltage (v_(aux)) at the terminals ofan auxiliary turn (22) coupled to the magnetic circuit (18) andconcatenating the magnetic flux (φ) is measured; the auxiliary turn (22)being substantially open electrically; and the time derivative of themagnetic flux (φ) and the magnetic flux (φ) itself being calculated byapplying the following formulae: $\begin{matrix}{\frac{{\phi (t)}}{t} = {v_{aus}(t)}} \\{{\phi (T)} = {{\int_{0}^{T}{{v_{aus}(t)}{t}}} + {\phi (0)}}}\end{matrix}$

in which φ is the magnetic flux (φ) and v_(aux) is the voltage (v_(aux))present at the terminals of the auxiliary turn (22).